The English Courts generally will not put time at large because of a failure to certify extensions of time unless there are special circumstances. The Courts approach is that if the CA fails to certify then the court itself (if asked to perform that task) can step into the shoes of the CA and operate the extension of time mechanisms properly and so the contractual provisions do not fall away.
The principle by which time becomes at large was recently commented on by Ramsey J in Bluewater Energy Services BV v Mercon Steel Structures BV and others:
“The principle is of some antiquity and has a surprising effect on the contractual obligations as to the time for completion. As I have found that there is an extension of time mechanism for acts of prevention and I am able… to determine the appropriate adjustments to the… Key Dates, this is not the opportunity to consider the underlying basis for the principle.”
You are therefore unlikely to succeed on this issue. Even if you did you would still have to complete the Works within a reasonable period of time and (if the LAD clause was found to be inoperable) then pay damages at large which could of course be greater than the level of LADs.
I suggest that you operate the mechanism for extensions of time, support your claim with expert evidence and then Adjudicate the issue. Good luck.